To gain qualitative insights, we focus on \(I^{(2)}_{A:B|M}\) , which can be studied analytically via a mapping to the effective Hamiltonian.
The essence of the mapping is to identify the average second moments \(\overline{\operatorname{tr}\tilde{\rho }_{A,m}^2}\)  and \(\overline{\operatorname{tr}\tilde{\rho }_{m}^2}\)  in Eq. (REF ) with the partition function of a classical spin model with certain boundary conditions.
To establish the mapping, we formulate the dynamics of the average purity, which involves two copies of density matrix, as the evolution of state vector \(\mbox{$| \rho  \rangle \rangle $} \equiv \rho \otimes \rho \)  in the duplicated Hilbert space \(\mathcal {H}^{(2)} = (\mathcal {H} \otimes \mathcal {H}^*)^{\otimes 2}\) .
Computing the subsystem purity is given by the overlap between \(\mbox{$| \rho  \rangle \rangle $}\)  and a reference state, which will manifest as boundary conditions in the classical spin model at final times [1]}, [2]}.
